Atmospheric neutrinos with three flavor mixing

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We analyze the atmospheric neutrino data in the context of three flavor neutrino oscillations taking account of the matter effects in the earth. With the hierarchy among the vacuum mass eigenvalues μ₃² ≫ μ₂² ≥ μ₁², the solution of the atmospheric neutrino problem depends on δ₃₁=μ₃² - μ₁² and the 13 and 23 mixing angles ϕ and ψ. Whereas the sub-GeV atmospheric neutrino data imposes only a lower limit on δ₃₁ > 10⁻³ eV², the zenith angle dependent suppression observed in the multi-GeV data limits δ₃₁ from above also. The allowed regions of the parameter space are strongly constrained by the multi-GeV data. Combined with our earlier solution to the solar neutrino problem which depends on δ₂₁= μ₂²-μ₁² and the 12 and 13 mixing angles ω and ϕ, we have obtained the ranges of values of the five neutrino parameters which solve both the solar and the atmospheric neutrino problems simultaneously.